Reliability and integrity measures of GPS positioning via geometrical constraints

Confidence domains for Global Navigation Satellite System (GNSS) positioning and inconsistency measures of the observations are of great importance for any navigation system, especially for safety critical applications. In this work, deterministic error bounds are derived from a sensitivity analysis of the observation correction models and introduced in form of intervals to assess remaining observation errors. Using convex optimization, bounding zones are computed for GPS positioning using geometrical constraints imposed by the observation intervals. The bounding zone is a convex polytope. We show that the volume of the polytope is an inconsistency measures rather than confidence measures, since small polytopes indicates bad consistency of the observations. In extreme cases empty sets are obtained which indicates large outliers. We explain how the shape and the volume of the polytope are related to the positioning geometry and how observations of maximum impact can be revealed. In a first attempt, a point position can by associated to the solution set by its barycenter. However, we show that this assignment has based to be interpreted with care. Furthermore, we propose a new concept of Minimum Detectable Biases (MDB) on the geometric relations. Taking GPS data from simulations and real experiments, a comparison analysis between the proposed deterministic bounding method and the classical least-squares adjustment has been conduct in terms of accuracy and reliability. This helps validating that our proposed deterministic bound methods shows high internal and external reliability compared to the probabilistic approaches and that it provides rigorous inconsistency measures.